url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
---|---|---|---|---|---|---|---|---|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | norm_cast | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - β(normSq u)) * deriv g 0
β’ Complex.abs (1 - β(normSq u)) β€ 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - β(normSq u)) * deriv g 0
β’ |1 - normSq u| β€ 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rw [abs_sub_le_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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - β(normSq u)) * deriv g 0
β’ |1 - normSq u| β€ 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - β(normSq u)) * deriv g 0
β’ 1 - normSq u β€ 1 β§ normSq u - 1 β€ 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | refine β¨?_, ?_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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - β(normSq u)) * deriv g 0
β’ 1 - normSq u β€ 1 β§ normSq u - 1 β€ 1 | case refine_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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - β(normSq u)) * deriv g 0
β’ 1 - normSq u β€ 1
case refine_2
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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - β(normSq u)) * deriv g 0
β’ normSq u - 1 β€ 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | repeat linarith | case refine_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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - β(normSq u)) * deriv g 0
β’ 1 - normSq u β€ 1
case refine_2
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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - β(normSq u)) * deriv g 0
β’ normSq u - 1 β€ 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
β’ 1 - β(normSq u) β 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | set w := 1 - conj u * u with hw | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
β’ deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u)) | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
w : β := 1 - (starRingEnd β) u * u
hw : w = 1 - (starRingEnd β) u * u
β’ deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have : w β 0 := by simpa [normSq_eq_conj_mul_self, mul_comm u] using e1 | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
w : β := 1 - (starRingEnd β) u * u
hw : w = 1 - (starRingEnd β) u * u
β’ deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u)) | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
w : β := 1 - (starRingEnd β) u * u
hw : w = 1 - (starRingEnd β) u * u
this : w β 0
β’ deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rw [Ο_deriv u_in_π» u_in_π», normSq_eq_conj_mul_self, mul_comm u, β hw] | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
w : β := 1 - (starRingEnd β) u * u
hw : w = 1 - (starRingEnd β) u * u
this : w β 0
β’ deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u)) | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
w : β := 1 - (starRingEnd β) u * u
hw : w = 1 - (starRingEnd β) u * u
this : w β 0
β’ w / w ^ 2 = 1 / w |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | field_simp | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
w : β := 1 - (starRingEnd β) u * u
hw : w = 1 - (starRingEnd β) u * u
this : w β 0
β’ w / w ^ 2 = 1 / w | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
w : β := 1 - (starRingEnd β) u * u
hw : w = 1 - (starRingEnd β) u * u
this : w β 0
β’ w * w = w ^ 2 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
w : β := 1 - (starRingEnd β) u * u
hw : w = 1 - (starRingEnd β) u * u
this : w β 0
β’ w * w = w ^ 2 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simpa [normSq_eq_conj_mul_self, mul_comm u] using e1 | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
w : β := 1 - (starRingEnd β) u * u
hw : w = 1 - (starRingEnd β) u * u
β’ w β 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rw [normSq_eq_abs] | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
β’ normSq u < 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
β’ Complex.abs u ^ 2 < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have : abs u < 1 := mem_π»_iff.mp 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
β’ Complex.abs u ^ 2 < 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
this : Complex.abs u < 1
β’ Complex.abs u ^ 2 < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | 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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (Ο u_in_π»).to_fun u * deriv f 0
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
this : Complex.abs u < 1
β’ Complex.abs u ^ 2 < 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | linarith | case refine_2
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 : Complex.abs (deriv g 0) β€ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - β(normSq u) β 0
Ο'u_u : deriv (Ο u_in_π»).to_fun u = 1 / (1 - β(normSq u))
e2 : 0 β€ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - β(normSq u)) * deriv g 0
β’ normSq u - 1 β€ 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | obtain β¨u, u_in_π», u_not_in_f_Uβ© := exists_of_ssubset hf | z u zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | let Οα΅€ : embedding π» π» := Ο u_in_π» | case intro.intro
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | let Οα΅€f : embedding U π» := Οα΅€.comp f | case intro.intro
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have Οα΅€f_ne_zero : β z β U, Οα΅€f z β 0 := Ξ» z z_in_U hz => by
refine u_not_in_f_U β¨z, z_in_U, ?_β©
apply Οα΅€.is_inj (f.maps_to z_in_U) u_in_π»
dsimp [Οα΅€f] at hz
rw [hz]
simp [Οα΅€, Ο] | case intro.intro
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | obtain β¨g, hgβ© := Οα΅€f.sqrt' Οα΅€f_ne_zero | case intro.intro
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | let v : β := g zβ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have v_in_π» : v β π» := g.maps_to hzβ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | let h : embedding U π» := (Ο v_in_π»).comp g | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have h_zβ_eq_0 : h zβ = 0 := by simp [h, Ο] | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | let Ο : β β β := Ξ» z => z ^ 2 | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | let Ο : β β β := Ο (neg_in_π» u_in_π») β Ο β Ο (neg_in_π» v_in_π») | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have f_eq_Ο_h : EqOn f (Ο β h) U := Ξ» z hz => by
have e1 := Ο_inv v_in_π» (g.maps_to hz)
have e2 := hg hz
have e3 := Ο_inv u_in_π» (f.maps_to hz)
dsimp [Οα΅€f] at e2
simp [Ο, Ο, h, e1, β e2, e3] | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have deriv_eq_mul : deriv f zβ = deriv Ο 0 * deriv h zβ := by
have e1 : U β π zβ := good_domain.is_open.mem_nhds hzβ
have e2 : π» β π (0 : β) := ball_mem_nhds _ zero_lt_one
have e3 : deriv f zβ = deriv (Ο β h) zβ := (eventuallyEq_of_mem e1 f_eq_Ο_h).deriv_eq
rw [e3, β h_zβ_eq_0]
refine deriv.comp zβ ?_ (h.is_diff.differentiableAt e1)
rw [h_zβ_eq_0]
exact Ο_is_diff.differentiableAt e2 | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | rw [deriv_eq_mul, norm_mul] | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ β h, βderiv f.to_fun zββ < βderiv h.to_fun zββ | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ β h_1, βderiv Ο 0β * βderiv h.to_fun zββ < βderiv h_1.to_fun zββ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | refine β¨h, mul_lt_of_lt_one_left ?_ ?_β© | case intro.intro.mk
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ β h_1, βderiv Ο 0β * βderiv h.to_fun zββ < βderiv h_1.to_fun zββ | case intro.intro.mk.refine_1
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ 0 < βderiv h.to_fun zββ
case intro.intro.mk.refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ βderiv Ο 0β < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | refine u_not_in_f_U β¨z, z_in_U, ?_β© | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
z : β
z_in_U : z β U
hz : Οα΅€f.to_fun z = 0
β’ False | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
z : β
z_in_U : z β U
hz : Οα΅€f.to_fun z = 0
β’ f.to_fun z = u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | apply Οα΅€.is_inj (f.maps_to z_in_U) u_in_π» | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
z : β
z_in_U : z β U
hz : Οα΅€f.to_fun z = 0
β’ f.to_fun z = u | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
z : β
z_in_U : z β U
hz : Οα΅€f.to_fun z = 0
β’ Οα΅€.to_fun (f.to_fun z) = Οα΅€.to_fun u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | dsimp [Οα΅€f] at hz | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
z : β
z_in_U : z β U
hz : Οα΅€f.to_fun z = 0
β’ Οα΅€.to_fun (f.to_fun z) = Οα΅€.to_fun u | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
z : β
z_in_U : z β U
hz : Οα΅€.to_fun (f.to_fun z) = 0
β’ Οα΅€.to_fun (f.to_fun z) = Οα΅€.to_fun u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | rw [hz] | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
z : β
z_in_U : z β U
hz : Οα΅€.to_fun (f.to_fun z) = 0
β’ Οα΅€.to_fun (f.to_fun z) = Οα΅€.to_fun u | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
z : β
z_in_U : z β U
hz : Οα΅€.to_fun (f.to_fun z) = 0
β’ 0 = Οα΅€.to_fun u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | simp [Οα΅€, Ο] | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
z : β
z_in_U : z β U
hz : Οα΅€.to_fun (f.to_fun z) = 0
β’ 0 = Οα΅€.to_fun u | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | simp [h, Ο] | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
β’ h.to_fun zβ = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have e1 := Ο_inv v_in_π» (g.maps_to hz) | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
z : β
hz : z β U
β’ f.to_fun z = (Ο β h.to_fun) z | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
z : β
hz : z β U
e1 : (Ο β―).to_fun ((Ο v_in_π»).to_fun (g.to_fun z)) = g.to_fun z
β’ f.to_fun z = (Ο β h.to_fun) z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have e2 := hg hz | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
z : β
hz : z β U
e1 : (Ο β―).to_fun ((Ο v_in_π»).to_fun (g.to_fun z)) = g.to_fun z
β’ f.to_fun z = (Ο β h.to_fun) z | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
z : β
hz : z β U
e1 : (Ο β―).to_fun ((Ο v_in_π»).to_fun (g.to_fun z)) = g.to_fun z
e2 : Οα΅€f.to_fun z = (g.to_fun ^ 2) z
β’ f.to_fun z = (Ο β h.to_fun) z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have e3 := Ο_inv u_in_π» (f.maps_to hz) | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
z : β
hz : z β U
e1 : (Ο β―).to_fun ((Ο v_in_π»).to_fun (g.to_fun z)) = g.to_fun z
e2 : Οα΅€f.to_fun z = (g.to_fun ^ 2) z
β’ f.to_fun z = (Ο β h.to_fun) z | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
z : β
hz : z β U
e1 : (Ο β―).to_fun ((Ο v_in_π»).to_fun (g.to_fun z)) = g.to_fun z
e2 : Οα΅€f.to_fun z = (g.to_fun ^ 2) z
e3 : (Ο β―).to_fun ((Ο u_in_π»).to_fun (f.to_fun z)) = f.to_fun z
β’ f.to_fun z = (Ο β h.to_fun) z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | dsimp [Οα΅€f] at e2 | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
z : β
hz : z β U
e1 : (Ο β―).to_fun ((Ο v_in_π»).to_fun (g.to_fun z)) = g.to_fun z
e2 : Οα΅€f.to_fun z = (g.to_fun ^ 2) z
e3 : (Ο β―).to_fun ((Ο u_in_π»).to_fun (f.to_fun z)) = f.to_fun z
β’ f.to_fun z = (Ο β h.to_fun) z | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
z : β
hz : z β U
e1 : (Ο β―).to_fun ((Ο v_in_π»).to_fun (g.to_fun z)) = g.to_fun z
e2 : Οα΅€.to_fun (f.to_fun z) = g.to_fun z ^ 2
e3 : (Ο β―).to_fun ((Ο u_in_π»).to_fun (f.to_fun z)) = f.to_fun z
β’ f.to_fun z = (Ο β h.to_fun) z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | simp [Ο, Ο, h, e1, β e2, e3] | zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
z : β
hz : z β U
e1 : (Ο β―).to_fun ((Ο v_in_π»).to_fun (g.to_fun z)) = g.to_fun z
e2 : Οα΅€.to_fun (f.to_fun z) = g.to_fun z ^ 2
e3 : (Ο β―).to_fun ((Ο u_in_π»).to_fun (f.to_fun z)) = f.to_fun z
β’ f.to_fun z = (Ο β h.to_fun) z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | refine (Ο (neg_in_π» u_in_π»)).is_diff.comp ?_ ?_ | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β Ο π» | case refine_1
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β (Ο β (Ο β―).to_fun) π»
case refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ MapsTo (Ο β (Ο β―).to_fun) π» π» |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | apply DifferentiableOn.comp | case refine_1
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β (Ο β (Ο β―).to_fun) π» | case refine_1.hg
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β Ο ?refine_1.t
case refine_1.hf
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β (Ο β―).to_fun π»
case refine_1.st
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ MapsTo (Ο β―).to_fun π» ?refine_1.t
case refine_1.t
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ Set β |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | case t => exact π» | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ Set β | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | case hg =>
apply DifferentiableOn.pow
exact differentiable_id.differentiableOn | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β Ο π» | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | case hf =>
exact (Ο (neg_in_π» v_in_π»)).is_diff | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β (Ο β―).to_fun π» | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | case st =>
exact (Ο (neg_in_π» v_in_π»)).maps_to | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ MapsTo (Ο β―).to_fun π» π» | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | exact π» | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ Set β | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | apply DifferentiableOn.pow | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β Ο π» | case ha
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β (fun x => x) π» |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | exact differentiable_id.differentiableOn | case ha
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β (fun x => x) π» | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | exact (Ο (neg_in_π» v_in_π»)).is_diff | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ DifferentiableOn β (Ο β―).to_fun π» | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | exact (Ο (neg_in_π» v_in_π»)).maps_to | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ MapsTo (Ο β―).to_fun π» π» | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | refine MapsTo.comp ?_ (Ο (neg_in_π» v_in_π»)).maps_to | case refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ MapsTo (Ο β (Ο β―).to_fun) π» π» | case refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ MapsTo Ο π» π» |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | intros z hz | case refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
β’ MapsTo Ο π» π» | case refine_2
zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
z : β
hz : z β π»
β’ Ο z β π» |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | simpa [Ο, π»] using hz | case refine_2
zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
z : β
hz : z β π»
β’ Ο z β π» | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have e1 : U β π zβ := good_domain.is_open.mem_nhds hzβ | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
β’ deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
β’ deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have e2 : π» β π (0 : β) := ball_mem_nhds _ zero_lt_one | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
β’ deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
e2 : π» β π 0
β’ deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have e3 : deriv f zβ = deriv (Ο β h) zβ := (eventuallyEq_of_mem e1 f_eq_Ο_h).deriv_eq | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
e2 : π» β π 0
β’ deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
e2 : π» β π 0
e3 : deriv f.to_fun zβ = deriv (Ο β h.to_fun) zβ
β’ deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | rw [e3, β h_zβ_eq_0] | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
e2 : π» β π 0
e3 : deriv f.to_fun zβ = deriv (Ο β h.to_fun) zβ
β’ deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
e2 : π» β π 0
e3 : deriv f.to_fun zβ = deriv (Ο β h.to_fun) zβ
β’ deriv (Ο β h.to_fun) zβ = deriv Ο (h.to_fun zβ) * deriv h.to_fun zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | refine deriv.comp zβ ?_ (h.is_diff.differentiableAt e1) | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
e2 : π» β π 0
e3 : deriv f.to_fun zβ = deriv (Ο β h.to_fun) zβ
β’ deriv (Ο β h.to_fun) zβ = deriv Ο (h.to_fun zβ) * deriv h.to_fun zβ | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
e2 : π» β π 0
e3 : deriv f.to_fun zβ = deriv (Ο β h.to_fun) zβ
β’ DifferentiableAt β Ο (h.to_fun zβ) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | rw [h_zβ_eq_0] | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
e2 : π» β π 0
e3 : deriv f.to_fun zβ = deriv (Ο β h.to_fun) zβ
β’ DifferentiableAt β Ο (h.to_fun zβ) | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
e2 : π» β π 0
e3 : deriv f.to_fun zβ = deriv (Ο β h.to_fun) zβ
β’ DifferentiableAt β Ο 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | exact Ο_is_diff.differentiableAt e2 | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
e1 : U β π zβ
e2 : π» β π 0
e3 : deriv f.to_fun zβ = deriv (Ο β h.to_fun) zβ
β’ DifferentiableAt β Ο 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | exact norm_pos_iff.2 (embedding.deriv_ne_zero good_domain.is_open hzβ) | case intro.intro.mk.refine_1
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ 0 < βderiv h.to_fun zββ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | apply non_injective_schwarz Ο_is_diff | case intro.intro.mk.refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ βderiv Ο 0β < 1 | case intro.intro.mk.refine_2.f_img
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ MapsTo Ο π» π»
case intro.intro.mk.refine_2.f_noninj
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ Β¬InjOn Ο π» |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | refine Ξ» z hz => (Ο (neg_in_π» u_in_π»)).maps_to (mem_π»_iff.mpr ?_) | case intro.intro.mk.refine_2.f_img
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ MapsTo Ο π» π» | case intro.intro.mk.refine_2.f_img
zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
z : β
hz : z β π»
β’ β(Ο β (Ο β―).to_fun) zβ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | simpa [Ο] using mem_π»_iff.mp ((Ο (neg_in_π» v_in_π»)).maps_to hz) | case intro.intro.mk.refine_2.f_img
zβ uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
z : β
hz : z β π»
β’ β(Ο β (Ο β―).to_fun) zβ < 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | simp only [InjOn, not_forall, exists_prop] | case intro.intro.mk.refine_2.f_noninj
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ Β¬InjOn Ο π» | case intro.intro.mk.refine_2.f_noninj
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ β x β π», β x_1 β π», Ο x = Ο x_1 β§ Β¬x = x_1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have e1 : (2β»ΒΉ : β) β π» := by apply mem_π»_iff.mpr; norm_num | case intro.intro.mk.refine_2.f_noninj
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ β x β π», β x_1 β π», Ο x = Ο x_1 β§ Β¬x = x_1 | case intro.intro.mk.refine_2.f_noninj
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
e1 : 2β»ΒΉ β π»
β’ β x β π», β x_1 β π», Ο x = Ο x_1 β§ Β¬x = x_1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have e2 : (-2β»ΒΉ : β) β π» := neg_in_π» e1 | case intro.intro.mk.refine_2.f_noninj
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
e1 : 2β»ΒΉ β π»
β’ β x β π», β x_1 β π», Ο x = Ο x_1 β§ Β¬x = x_1 | case intro.intro.mk.refine_2.f_noninj
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
β’ β x β π», β x_1 β π», Ο x = Ο x_1 β§ Β¬x = x_1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | refine β¨Ο v_in_π» 2β»ΒΉ, (Ο v_in_π»).maps_to e1, Ο v_in_π» (-2β»ΒΉ), (Ο v_in_π»).maps_to e2, ?_, ?_β© | case intro.intro.mk.refine_2.f_noninj
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
β’ β x β π», β x_1 β π», Ο x = Ο x_1 β§ Β¬x = x_1 | case intro.intro.mk.refine_2.f_noninj.refine_1
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
β’ Ο ((Ο v_in_π»).to_fun 2β»ΒΉ) = Ο ((Ο v_in_π»).to_fun (-2β»ΒΉ))
case intro.intro.mk.refine_2.f_noninj.refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
β’ Β¬(Ο v_in_π»).to_fun 2β»ΒΉ = (Ο v_in_π»).to_fun (-2β»ΒΉ) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | apply mem_π»_iff.mpr | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ 2β»ΒΉ β π» | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ β2β»ΒΉβ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | norm_num | z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
β’ β2β»ΒΉβ < 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | unfold_let | case intro.intro.mk.refine_2.f_noninj.refine_1
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
β’ Ο ((Ο v_in_π»).to_fun 2β»ΒΉ) = Ο ((Ο v_in_π»).to_fun (-2β»ΒΉ)) | case intro.intro.mk.refine_2.f_noninj.refine_1
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
β’ ((Ο β―).to_fun β (fun z => z ^ 2) β (Ο β―).to_fun) ((Ο v_in_π»).to_fun 2β»ΒΉ) =
((Ο β―).to_fun β (fun z => z ^ 2) β (Ο β―).to_fun) ((Ο v_in_π»).to_fun (-2β»ΒΉ)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | simp [Ο_inv v_in_π» e1, Ο_inv v_in_π» e2] | case intro.intro.mk.refine_2.f_noninj.refine_1
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
β’ ((Ο β―).to_fun β (fun z => z ^ 2) β (Ο β―).to_fun) ((Ο v_in_π»).to_fun 2β»ΒΉ) =
((Ο β―).to_fun β (fun z => z ^ 2) β (Ο β―).to_fun) ((Ο v_in_π»).to_fun (-2β»ΒΉ)) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | intro h | case intro.intro.mk.refine_2.f_noninj.refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
h : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : h.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β h.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv h.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
β’ Β¬(Ο v_in_π»).to_fun 2β»ΒΉ = (Ο v_in_π»).to_fun (-2β»ΒΉ) | case intro.intro.mk.refine_2.f_noninj.refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
hβ : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : hβ.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β hβ.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv hβ.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
h : (Ο v_in_π»).to_fun 2β»ΒΉ = (Ο v_in_π»).to_fun (-2β»ΒΉ)
β’ False |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | have := (Ο v_in_π»).is_inj e1 e2 h | case intro.intro.mk.refine_2.f_noninj.refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
hβ : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : hβ.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β hβ.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv hβ.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
h : (Ο v_in_π»).to_fun 2β»ΒΉ = (Ο v_in_π»).to_fun (-2β»ΒΉ)
β’ False | case intro.intro.mk.refine_2.f_noninj.refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
hβ : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : hβ.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β hβ.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv hβ.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
h : (Ο v_in_π»).to_fun 2β»ΒΉ = (Ο v_in_π»).to_fun (-2β»ΒΉ)
this : 2β»ΒΉ = -2β»ΒΉ
β’ False |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | step_2 | [114, 1] | [174, 25] | norm_num at this | case intro.intro.mk.refine_2.f_noninj.refine_2
z uβ zβ : β
U : Set β
instβ : good_domain U
hzβ : zβ β U
f : embedding U π»
hf : f.to_fun '' U β π»
u : β
u_in_π» : u β π»
u_not_in_f_U : u β f.to_fun '' U
Οα΅€ : embedding π» π» := Ο u_in_π»
Οα΅€f : embedding U π» := embedding.comp Οα΅€ f
Οα΅€f_ne_zero : β z β U, Οα΅€f.to_fun z β 0
g : embedding U π»
hg : EqOn Οα΅€f.to_fun (g.to_fun ^ 2) U
v : β := g.to_fun zβ
v_in_π» : v β π»
hβ : embedding U π» := embedding.comp (Ο v_in_π») g
h_zβ_eq_0 : hβ.to_fun zβ = 0
Ο : β β β := fun z => z ^ 2
Ο : β β β := (Ο β―).to_fun β Ο β (Ο β―).to_fun
f_eq_Ο_h : EqOn f.to_fun (Ο β hβ.to_fun) U
Ο_is_diff : DifferentiableOn β Ο π»
deriv_eq_mul : deriv f.to_fun zβ = deriv Ο 0 * deriv hβ.to_fun zβ
e1 : 2β»ΒΉ β π»
e2 : -2β»ΒΉ β π»
h : (Ο v_in_π»).to_fun 2β»ΒΉ = (Ο v_in_π»).to_fun (-2β»ΒΉ)
this : 2β»ΒΉ = -2β»ΒΉ
β’ False | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | isCompact_segment | [8, 1] | [12, 74] | simpa only [segment_eq_image] using isCompact_Icc.image (by continuity) | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
n : β
instββΈ : OrderedRing π
instββ· : TopologicalSpace π
instββΆ : TopologicalAddGroup π
instββ΅ : CompactIccSpace π
instββ΄ : TopologicalSpace E
instβΒ³ : AddCommGroup E
instβΒ² : ContinuousAdd E
instβΒΉ : Module π E
instβ : ContinuousSMul π E
x y : E
β’ IsCompact (segment π x y) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | isCompact_segment | [8, 1] | [12, 74] | continuity | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
n : β
instββΈ : OrderedRing π
instββ· : TopologicalSpace π
instββΆ : TopologicalAddGroup π
instββ΅ : CompactIccSpace π
instββ΄ : TopologicalSpace E
instβΒ³ : AddCommGroup E
instβΒ² : ContinuousAdd E
instβΒΉ : Module π E
instβ : ContinuousSMul π E
x y : E
β’ Continuous fun ΞΈ => (1 - ΞΈ) β’ x + ΞΈ β’ y | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | mem_closed_ball_neg_iff_mem_neg_closed_ball | [14, 1] | [16, 33] | rw [β neg_closedBall r v] | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
n : β
instβ : SeminormedAddCommGroup V
u v : V
β’ u β closedBall (-v) r β -u β closedBall v r | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
n : β
instβ : SeminormedAddCommGroup V
u v : V
β’ u β -closedBall v r β -u β closedBall v r |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | mem_closed_ball_neg_iff_mem_neg_closed_ball | [14, 1] | [16, 33] | rfl | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
n : β
instβ : SeminormedAddCommGroup V
u v : V
β’ u β -closedBall v r β -u β closedBall v r | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | DifferentiableAt.deriv_eq_deriv_pow_div_pow | [18, 1] | [24, 7] | have h1 : g z β 0 := Ξ» h => fz_nonzero (by simp [Eventually.self_of_nhds hg, h, n_pos.ne.symm]) | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
nβ n : β
n_pos : 0 < n
f g : β β β
hg : βαΆ (z : β) in π z, f z = g z ^ n
g_diff : DifferentiableAt β g z
fz_nonzero : f z β 0
β’ deriv g z = deriv f z / (βn * g z ^ (n - 1)) | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
nβ n : β
n_pos : 0 < n
f g : β β β
hg : βαΆ (z : β) in π z, f z = g z ^ n
g_diff : DifferentiableAt β g z
fz_nonzero : f z β 0
h1 : g z β 0
β’ deriv g z = deriv f z / (βn * g z ^ (n - 1)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | DifferentiableAt.deriv_eq_deriv_pow_div_pow | [18, 1] | [24, 7] | have h2 : n * (g z) ^ (n - 1) β 0 := by simp [pow_ne_zero, h1, n_pos.ne.symm] | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
nβ n : β
n_pos : 0 < n
f g : β β β
hg : βαΆ (z : β) in π z, f z = g z ^ n
g_diff : DifferentiableAt β g z
fz_nonzero : f z β 0
h1 : g z β 0
β’ deriv g z = deriv f z / (βn * g z ^ (n - 1)) | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
nβ n : β
n_pos : 0 < n
f g : β β β
hg : βαΆ (z : β) in π z, f z = g z ^ n
g_diff : DifferentiableAt β g z
fz_nonzero : f z β 0
h1 : g z β 0
h2 : βn * g z ^ (n - 1) β 0
β’ deriv g z = deriv f z / (βn * g z ^ (n - 1)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | DifferentiableAt.deriv_eq_deriv_pow_div_pow | [18, 1] | [24, 7] | rw [(EventuallyEq.deriv hg).self_of_nhds, deriv_pow'' _ g_diff, eq_div_iff h2] | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
nβ n : β
n_pos : 0 < n
f g : β β β
hg : βαΆ (z : β) in π z, f z = g z ^ n
g_diff : DifferentiableAt β g z
fz_nonzero : f z β 0
h1 : g z β 0
h2 : βn * g z ^ (n - 1) β 0
β’ deriv g z = deriv f z / (βn * g z ^ (n - 1)) | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
nβ n : β
n_pos : 0 < n
f g : β β β
hg : βαΆ (z : β) in π z, f z = g z ^ n
g_diff : DifferentiableAt β g z
fz_nonzero : f z β 0
h1 : g z β 0
h2 : βn * g z ^ (n - 1) β 0
β’ deriv g z * (βn * g z ^ (n - 1)) = βn * g z ^ (n - 1) * deriv g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | DifferentiableAt.deriv_eq_deriv_pow_div_pow | [18, 1] | [24, 7] | ring | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
nβ n : β
n_pos : 0 < n
f g : β β β
hg : βαΆ (z : β) in π z, f z = g z ^ n
g_diff : DifferentiableAt β g z
fz_nonzero : f z β 0
h1 : g z β 0
h2 : βn * g z ^ (n - 1) β 0
β’ deriv g z * (βn * g z ^ (n - 1)) = βn * g z ^ (n - 1) * deriv g z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | DifferentiableAt.deriv_eq_deriv_pow_div_pow | [18, 1] | [24, 7] | simp [Eventually.self_of_nhds hg, h, n_pos.ne.symm] | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
nβ n : β
n_pos : 0 < n
f g : β β β
hg : βαΆ (z : β) in π z, f z = g z ^ n
g_diff : DifferentiableAt β g z
fz_nonzero : f z β 0
h : g z = 0
β’ f z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | DifferentiableAt.deriv_eq_deriv_pow_div_pow | [18, 1] | [24, 7] | simp [pow_ne_zero, h1, n_pos.ne.symm] | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
nβ n : β
n_pos : 0 < n
f g : β β β
hg : βαΆ (z : β) in π z, f z = g z ^ n
g_diff : DifferentiableAt β g z
fz_nonzero : f z β 0
h1 : g z β 0
β’ βn * g z ^ (n - 1) β 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | has_deriv_at_integral_of_continuous_of_lip | [30, 1] | [50, 80] | simp only [intervalIntegral, not_lt, hab, Ioc_eq_empty, Measure.restrict_empty,
integral_zero_measure, sub_zero] | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
β’ HasDerivAt (fun z => β« (t : β) in a..b, Ο z t) (β« (t : β) in a..b, Ο t) zβ | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | has_deriv_at_integral_of_continuous_of_lip | [30, 1] | [50, 80] | let ΞΌ : Measure β := volume.restrict (Ioc a b) | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | has_deriv_at_integral_of_continuous_of_lip | [30, 1] | [50, 80] | have h1 : βαΆ z in π zβ, AEStronglyMeasurable (Ο z) ΞΌ :=
Ο_cts.mono (Ξ» z h => (h.mono Ioc_subset_Icc_self).aestronglyMeasurable measurableSet_Ioc) | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | has_deriv_at_integral_of_continuous_of_lip | [30, 1] | [50, 80] | have h2 : Integrable (Ο zβ) ΞΌ :=
Ο_cts.self_of_nhds.integrableOn_Icc.mono_set Ioc_subset_Icc_self | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
h2 : Integrable (Ο zβ) ΞΌ
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | has_deriv_at_integral_of_continuous_of_lip | [30, 1] | [50, 80] | have h3 : AEStronglyMeasurable Ο ΞΌ := Ο_cts.aestronglyMeasurable measurableSet_Ioc | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
h2 : Integrable (Ο zβ) ΞΌ
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
h2 : Integrable (Ο zβ) ΞΌ
h3 : AEStronglyMeasurable Ο ΞΌ
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | has_deriv_at_integral_of_continuous_of_lip | [30, 1] | [50, 80] | have h4 : βα΅ t βΞΌ, LipschitzOnWith (Real.nnabs C) (Ξ» z => Ο z t) (ball zβ Ξ΄) :=
(ae_restrict_iff' measurableSet_Ioc).mpr (eventually_of_forall Ο_lip) | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
h2 : Integrable (Ο zβ) ΞΌ
h3 : AEStronglyMeasurable Ο ΞΌ
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
h2 : Integrable (Ο zβ) ΞΌ
h3 : AEStronglyMeasurable Ο ΞΌ
h4 : βα΅ (t : β) βΞΌ, LipschitzOnWith (nnabs C) (fun z => Ο z t) (ball zβ Ξ΄)
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | has_deriv_at_integral_of_continuous_of_lip | [30, 1] | [50, 80] | have h5 : Integrable (Ξ» _ => C) ΞΌ := integrable_const _ | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
h2 : Integrable (Ο zβ) ΞΌ
h3 : AEStronglyMeasurable Ο ΞΌ
h4 : βα΅ (t : β) βΞΌ, LipschitzOnWith (nnabs C) (fun z => Ο z t) (ball zβ Ξ΄)
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
h2 : Integrable (Ο zβ) ΞΌ
h3 : AEStronglyMeasurable Ο ΞΌ
h4 : βα΅ (t : β) βΞΌ, LipschitzOnWith (nnabs C) (fun z => Ο z t) (ball zβ Ξ΄)
h5 : Integrable (fun x => C) ΞΌ
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | has_deriv_at_integral_of_continuous_of_lip | [30, 1] | [50, 80] | have h6 : βα΅ t βΞΌ, HasDerivAt (Ξ» z => Ο z t) (Ο t) zβ :=
(ae_restrict_iff' measurableSet_Ioc).mpr (eventually_of_forall Ο_der) | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
h2 : Integrable (Ο zβ) ΞΌ
h3 : AEStronglyMeasurable Ο ΞΌ
h4 : βα΅ (t : β) βΞΌ, LipschitzOnWith (nnabs C) (fun z => Ο z t) (ball zβ Ξ΄)
h5 : Integrable (fun x => C) ΞΌ
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
h2 : Integrable (Ο zβ) ΞΌ
h3 : AEStronglyMeasurable Ο ΞΌ
h4 : βα΅ (t : β) βΞΌ, LipschitzOnWith (nnabs C) (fun z => Ο z t) (ball zβ Ξ΄)
h5 : Integrable (fun x => C) ΞΌ
h6 : βα΅ (t : β) βΞΌ, HasDerivAt (fun z => Ο z t) (Ο t) zβ
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | has_deriv_at_integral_of_continuous_of_lip | [30, 1] | [50, 80] | exact (hasDerivAt_integral_of_dominated_loc_of_lip Ξ΄_pos h1 h2 h3 h4 h5 h6).2 | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
aβ bβ t : β
n : β
Ο : β β β β β
Ο : β β β
zβ : β
a b C Ξ΄ : β
hab : a β€ b
Ξ΄_pos : 0 < Ξ΄
Ο_cts : βαΆ (z : β) in π zβ, ContinuousOn (Ο z) (Icc a b)
Ο_der : β t β Ioc a b, HasDerivAt (fun x => Ο x t) (Ο t) zβ
Ο_lip : β t β Ioc a b, LipschitzOnWith (nnabs C) (fun x => Ο x t) (ball zβ Ξ΄)
Ο_cts : ContinuousOn Ο (Ioc a b)
ΞΌ : Measure β := Measure.restrict volume (Ioc a b)
h1 : βαΆ (z : β) in π zβ, AEStronglyMeasurable (Ο z) ΞΌ
h2 : Integrable (Ο zβ) ΞΌ
h3 : AEStronglyMeasurable Ο ΞΌ
h4 : βα΅ (t : β) βΞΌ, LipschitzOnWith (nnabs C) (fun z => Ο z t) (ball zβ Ξ΄)
h5 : Integrable (fun x => C) ΞΌ
h6 : βα΅ (t : β) βΞΌ, HasDerivAt (fun z => Ο z t) (Ο t) zβ
β’ HasDerivAt (fun z => β« (x : β) in Ioc a b, Ο z x βvolume) (β« (x : β) in Ioc a b, Ο x βvolume) zβ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | uIoo_eq_union | [56, 1] | [57, 40] | cases le_total a b <;> simp [*, uIoo] | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
n : β
β’ uIoo a b = Ioo a b βͺ Ioo b a | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | mem_uIoo | [59, 1] | [59, 93] | simp [uIoo_eq_union] | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
n : β
β’ t β uIoo a b β a < t β§ t < b β¨ b < t β§ t < a | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | uIoo_eq_uIoc_sdiff_ends | [61, 1] | [71, 36] | ext t | π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b t : β
n : β
β’ uIoo a b = Ξ a b \ {a, b} | case h
π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b tβ : β
n : β
t : β
β’ t β uIoo a b β t β Ξ a b \ {a, b} |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/to_mathlib.lean | uIoo_eq_uIoc_sdiff_ends | [61, 1] | [71, 36] | constructor <;> intro hh | case h
π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b tβ : β
n : β
t : β
β’ t β uIoo a b β t β Ξ a b \ {a, b} | case h.mp
π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b tβ : β
n : β
t : β
hh : t β uIoo a b
β’ t β Ξ a b \ {a, b}
case h.mpr
π : Type u_1
E : Type u_2
V : Type u_3
r : β
z : β
a b tβ : β
n : β
t : β
hh : t β Ξ a b \ {a, b}
β’ t β uIoo a b |
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