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
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stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
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stringlengths 6
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https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | ContinuousOn_uderiv | [30, 1] | [35, 27] | exact nhdsWithin_le_nhds | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
f : π U
β’ Tendsto (fun f => f) (π[π U] f) (π f) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_const | [42, 1] | [43, 28] | simp [π, β mapsTo_sUnion] | U : Set β
Qβ : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
Q : Set β
β’ (π U fun x => Q) = {f | f β π U β§ MapsTo f U Q} | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | isClosed_π | [45, 1] | [53, 91] | rw [π, setOf_and] | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ IsClosed (π U Q) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ IsClosed ({a | a β π U} β© {a | β K β compacts U, MapsTo a K (Q K)}) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | isClosed_π | [45, 1] | [53, 91] | apply (isClosed_π hU).inter | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ IsClosed ({a | a β π U} β© {a | β K β compacts U, MapsTo a K (Q K)}) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ IsClosed {a | β K β compacts U, MapsTo a K (Q K)} |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | isClosed_π | [45, 1] | [53, 91] | simp only [setOf_forall, MapsTo] | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ IsClosed {a | β K β compacts U, MapsTo a K (Q K)} | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ IsClosed (β i β compacts U, β i_1 β i, {x | x i_1 β Q i}) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | isClosed_π | [45, 1] | [53, 91] | apply isClosed_biInter | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ IsClosed (β i β compacts U, β i_1 β i, {x | x i_1 β Q i}) | case h
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ β i β compacts U, IsClosed (β i_1 β i, {x | x i_1 β Q i}) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | isClosed_π | [45, 1] | [53, 91] | intro K hK | case h
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ β i β compacts U, IsClosed (β i_1 β i, {x | x i_1 β Q i}) | case h
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
K : Set β
hK : K β compacts U
β’ IsClosed (β i β K, {x | x i β Q K}) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | isClosed_π | [45, 1] | [53, 91] | apply isClosed_biInter | case h
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
K : Set β
hK : K β compacts U
β’ IsClosed (β i β K, {x | x i β Q K}) | case h.h
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
K : Set β
hK : K β compacts U
β’ β i β K, IsClosed {x | x i β Q K} |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | isClosed_π | [45, 1] | [53, 91] | intro z hz | case h.h
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
K : Set β
hK : K β compacts U
β’ β i β K, IsClosed {x | x i β Q K} | case h.h
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
K : Set β
hK : K β compacts U
z : β
hz : z β K
β’ IsClosed {x | x z β Q K} |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | isClosed_π | [45, 1] | [53, 91] | apply (hQ K hK).isClosed.preimage | case h.h
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
K : Set β
hK : K β compacts U
z : β
hz : z β K
β’ IsClosed {x | x z β Q K} | case h.h
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
K : Set β
hK : K β compacts U
z : β
hz : z β K
β’ Continuous fun x => x z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | isClosed_π | [45, 1] | [53, 91] | exact ((UniformOnFun.uniformContinuous_eval_of_mem β (compacts U)
(mem_singleton z) β¨singleton_subset_iff.2 (hK.1 hz), isCompact_singletonβ©).continuous) | case h.h
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
K : Set β
hK : K β compacts U
z : β
hz : z β K
β’ Continuous fun x => x z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | IsClosed_π | [61, 1] | [67, 84] | suffices h : IsClosed {f : π U | MapsTo f U (closedBall 0 1)} by
exact (isClosed_π hU).inter h | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
β’ IsClosed (π U) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
β’ IsClosed {f | MapsTo f U (closedBall 0 1)} |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | IsClosed_π | [61, 1] | [67, 84] | simp_rw [MapsTo, setOf_forall] | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
β’ IsClosed {f | MapsTo f U (closedBall 0 1)} | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
β’ IsClosed (β i β U, {x | x i β closedBall 0 1}) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | IsClosed_π | [61, 1] | [67, 84] | refine isClosed_biInter (Ξ» z hz => isClosed_ball.preimage ?_) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
β’ IsClosed (β i β U, {x | x i β closedBall 0 1}) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
z : β
hz : z β U
β’ Continuous fun x => x z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | IsClosed_π | [61, 1] | [67, 84] | exact ((UniformOnFun.uniformContinuous_eval_of_mem β (compacts U)
(mem_singleton z) β¨singleton_subset_iff.2 hz, isCompact_singletonβ©).continuous) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
z : β
hz : z β U
β’ Continuous fun x => x z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | IsClosed_π | [61, 1] | [67, 84] | exact (isClosed_π hU).inter h | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
hU : IsOpen U
h : IsClosed {f | MapsTo f U (closedBall 0 1)}
β’ IsClosed (π U) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | obtain β¨u, huβ© := nonempty_compl.mpr (good_domain.ne_univ : U β univ) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
β’ Set.Nonempty (π U) | case intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | let f : β β β := Ξ» z => z - u | case intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
β’ Set.Nonempty (π U) | case intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have f_inj : Injective f := Ξ» _ _ h => sub_left_inj.mp h | case intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
β’ Set.Nonempty (π U) | case intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have f_hol : DifferentiableOn β f U := differentiableOn_id.sub (differentiableOn_const u) | case intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
β’ Set.Nonempty (π U) | case intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have f_noz : β β¦z : ββ¦, z β U -> f z β 0 := Ξ» z hz f0 => hu (sub_eq_zero.mp f0 βΈ hz) | case intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
β’ Set.Nonempty (π U) | case intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | obtain β¨g, g_hol, g_sqfβ© := good_domain.has_sqrt f f_noz f_hol | case intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
β’ Set.Nonempty (π U) | case intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | obtain β¨zβ, hzββ© := (good_domain.is_nonempty : U.Nonempty) | case intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
β’ Set.Nonempty (π U) | case intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have gU_nhd : g '' U β π (g zβ) := by
have e1 : U β π zβ := good_domain.is_open.mem_nhds hzβ
have e2 := g_hol.analyticAt e1
have f_eq_comp := (good_domain.is_open.eventually_mem hzβ).mono g_sqf
have dg_nonzero : deriv g zβ β 0 := by
have e3 := e2.differentiableAt.deriv_eq_deriv_pow_div_pow zero_lt_two f_eq_comp (f_noz hzβ)
simp [e3, deriv_sub_const, f]
intro h
have := g_sqf hzβ
rw [Pi.pow_apply, h, zero_pow two_ne_zero] at this
cases f_noz hzβ this
refine e2.eventually_constant_or_nhds_le_map_nhds.resolve_left (Ξ» h => ?_) (image_mem_map e1)
simp [EventuallyEq.deriv_eq h] at dg_nonzero | case intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
β’ Set.Nonempty (π U) | case intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | obtain β¨r, r_pos, hrβ© := Metric.mem_nhds_iff.mp gU_nhd | case intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
β’ Set.Nonempty (π U) | case intro.intro.intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | let gg : embedding U ((closedBall (- g zβ) (r / 2))αΆ) :=
{ to_fun := g,
is_diff := g_hol,
is_inj := Ξ» zβ hzβ zβ hzβ hgz => f_inj (by simp [g_sqf _, hzβ, hzβ, hgz]),
maps_to := Ξ» z hz hgz => by {
apply f_noz hz
rw [mem_closed_ball_neg_iff_mem_neg_closed_ball] at hgz
obtain β¨z', hz', hgz'β© := (closedBall_subset_ball (by linarith)).trans hr hgz
have hzz' : z = z' := f_inj (by simp [g_sqf hz, g_sqf hz', hgz'])
simpa [hzz', neg_eq_self_iff, g_sqf hz'] using hgz'.symm } } | case intro.intro.intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
β’ Set.Nonempty (π U) | case intro.intro.intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
gg : embedding U (closedBall (-g zβ) (r / 2))αΆ := { to_fun := g, is_diff := g_hol, is_inj := β―, maps_to := β― }
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | let ggg := (embedding.inv _ (by linarith)).comp gg | case intro.intro.intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
gg : embedding U (closedBall (-g zβ) (r / 2))αΆ := { to_fun := g, is_diff := g_hol, is_inj := β―, maps_to := β― }
β’ Set.Nonempty (π U) | case intro.intro.intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
gg : embedding U (closedBall (-g zβ) (r / 2))αΆ := { to_fun := g, is_diff := g_hol, is_inj := β―, maps_to := β― }
ggg : embedding U π» := embedding.comp (embedding.inv (-g zβ) β―) gg
β’ Set.Nonempty (π U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | refine β¨ggg.to_fun, β¨ggg.is_diff, ?_β©, ggg.is_injβ© | case intro.intro.intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
gg : embedding U (closedBall (-g zβ) (r / 2))αΆ := { to_fun := g, is_diff := g_hol, is_inj := β―, maps_to := β― }
ggg : embedding U π» := embedding.comp (embedding.inv (-g zβ) β―) gg
β’ Set.Nonempty (π U) | case intro.intro.intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
gg : embedding U (closedBall (-g zβ) (r / 2))αΆ := { to_fun := g, is_diff := g_hol, is_inj := β―, maps_to := β― }
ggg : embedding U π» := embedding.comp (embedding.inv (-g zβ) β―) gg
β’ MapsTo ggg.to_fun U (closedBall 0 1) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | exact Ξ» z hz => ball_subset_closedBall (ggg.maps_to hz) | case intro.intro.intro.intro.intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
gg : embedding U (closedBall (-g zβ) (r / 2))αΆ := { to_fun := g, is_diff := g_hol, is_inj := β―, maps_to := β― }
ggg : embedding U π» := embedding.comp (embedding.inv (-g zβ) β―) gg
β’ MapsTo ggg.to_fun U (closedBall 0 1) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have e1 : U β π zβ := good_domain.is_open.mem_nhds hzβ | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
β’ g '' U β π (g zβ) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
β’ g '' U β π (g zβ) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have e2 := g_hol.analyticAt e1 | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
β’ g '' U β π (g zβ) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
β’ g '' U β π (g zβ) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have f_eq_comp := (good_domain.is_open.eventually_mem hzβ).mono g_sqf | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
β’ g '' U β π (g zβ) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
β’ g '' U β π (g zβ) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have dg_nonzero : deriv g zβ β 0 := by
have e3 := e2.differentiableAt.deriv_eq_deriv_pow_div_pow zero_lt_two f_eq_comp (f_noz hzβ)
simp [e3, deriv_sub_const, f]
intro h
have := g_sqf hzβ
rw [Pi.pow_apply, h, zero_pow two_ne_zero] at this
cases f_noz hzβ this | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
β’ g '' U β π (g zβ) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
dg_nonzero : deriv g zβ β 0
β’ g '' U β π (g zβ) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | refine e2.eventually_constant_or_nhds_le_map_nhds.resolve_left (Ξ» h => ?_) (image_mem_map e1) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
dg_nonzero : deriv g zβ β 0
β’ g '' U β π (g zβ) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
dg_nonzero : deriv g zβ β 0
h : βαΆ (z : β) in π zβ, g z = g zβ
β’ False |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | simp [EventuallyEq.deriv_eq h] at dg_nonzero | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
dg_nonzero : deriv g zβ β 0
h : βαΆ (z : β) in π zβ, g z = g zβ
β’ False | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have e3 := e2.differentiableAt.deriv_eq_deriv_pow_div_pow zero_lt_two f_eq_comp (f_noz hzβ) | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
β’ deriv g zβ β 0 | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
e3 : deriv (fun z => g z) zβ = deriv (fun z => f z) zβ / (β2 * g zβ ^ (2 - 1))
β’ deriv g zβ β 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | simp [e3, deriv_sub_const, f] | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
e3 : deriv (fun z => g z) zβ = deriv (fun z => f z) zβ / (β2 * g zβ ^ (2 - 1))
β’ deriv g zβ β 0 | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
e3 : deriv (fun z => g z) zβ = deriv (fun z => f z) zβ / (β2 * g zβ ^ (2 - 1))
β’ Β¬g zβ = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | intro h | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
e3 : deriv (fun z => g z) zβ = deriv (fun z => f z) zβ / (β2 * g zβ ^ (2 - 1))
β’ Β¬g zβ = 0 | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
e3 : deriv (fun z => g z) zβ = deriv (fun z => f z) zβ / (β2 * g zβ ^ (2 - 1))
h : g zβ = 0
β’ False |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have := g_sqf hzβ | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
e3 : deriv (fun z => g z) zβ = deriv (fun z => f z) zβ / (β2 * g zβ ^ (2 - 1))
h : g zβ = 0
β’ False | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
e3 : deriv (fun z => g z) zβ = deriv (fun z => f z) zβ / (β2 * g zβ ^ (2 - 1))
h : g zβ = 0
this : f zβ = (g ^ 2) zβ
β’ False |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | rw [Pi.pow_apply, h, zero_pow two_ne_zero] at this | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
e3 : deriv (fun z => g z) zβ = deriv (fun z => f z) zβ / (β2 * g zβ ^ (2 - 1))
h : g zβ = 0
this : f zβ = (g ^ 2) zβ
β’ False | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
e3 : deriv (fun z => g z) zβ = deriv (fun z => f z) zβ / (β2 * g zβ ^ (2 - 1))
h : g zβ = 0
this : f zβ = 0
β’ False |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | cases f_noz hzβ this | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
e1 : U β π zβ
e2 : AnalyticAt β g zβ
f_eq_comp : βαΆ (x : β) in π zβ, f x = (g ^ 2) x
e3 : deriv (fun z => g z) zβ = deriv (fun z => f z) zβ / (β2 * g zβ ^ (2 - 1))
h : g zβ = 0
this : f zβ = 0
β’ False | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | simp [g_sqf _, hzβ, hzβ, hgz] | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
zβ : β
hzβ : zβ β U
zβ : β
hzβ : zβ β U
hgz : g zβ = g zβ
β’ f zβ = f zβ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | apply f_noz hz | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : g z β closedBall (-g zβ) (r / 2)
β’ False | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : g z β closedBall (-g zβ) (r / 2)
β’ f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | rw [mem_closed_ball_neg_iff_mem_neg_closed_ball] at hgz | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : g z β closedBall (-g zβ) (r / 2)
β’ f z = 0 | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : -g z β closedBall (g zβ) (r / 2)
β’ f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | obtain β¨z', hz', hgz'β© := (closedBall_subset_ball (by linarith)).trans hr hgz | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : -g z β closedBall (g zβ) (r / 2)
β’ f z = 0 | case intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : -g z β closedBall (g zβ) (r / 2)
z' : β
hz' : z' β U
hgz' : g z' = -g z
β’ f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | have hzz' : z = z' := f_inj (by simp [g_sqf hz, g_sqf hz', hgz']) | case intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : -g z β closedBall (g zβ) (r / 2)
z' : β
hz' : z' β U
hgz' : g z' = -g z
β’ f z = 0 | case intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : -g z β closedBall (g zβ) (r / 2)
z' : β
hz' : z' β U
hgz' : g z' = -g z
hzz' : z = z'
β’ f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | simpa [hzz', neg_eq_self_iff, g_sqf hz'] using hgz'.symm | case intro.intro
U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : -g z β closedBall (g zβ) (r / 2)
z' : β
hz' : z' β U
hgz' : g z' = -g z
hzz' : z = z'
β’ f z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | linarith | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : -g z β closedBall (g zβ) (r / 2)
β’ r / 2 < r | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | simp [g_sqf hz, g_sqf hz', hgz'] | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
z : β
hz : z β U
hgz : -g z β closedBall (g zβ) (r / 2)
z' : β
hz' : z' β U
hgz' : g z' = -g z
β’ f z = f z' | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Spaces.lean | π_nonempty | [73, 1] | [111, 58] | linarith | U : Set β
Q : Set β β Set β
ΞΉ : Type u_1
l : Filter ΞΉ
instβ : good_domain U
u : β
hu : u β UαΆ
f : β β β := fun z => z - u
f_inj : Injective f
f_hol : DifferentiableOn β f U
f_noz : β β¦z : ββ¦, z β U β f z β 0
g : β β β
g_hol : DifferentiableOn β g U
g_sqf : EqOn f (g ^ 2) U
zβ : β
hzβ : zβ β U
gU_nhd : g '' U β π (g zβ)
r : β
r_pos : r > 0
hr : ball (g zβ) r β g '' U
gg : embedding U (closedBall (-g zβ) (r / 2))αΆ := { to_fun := g, is_diff := g_hol, is_inj := β―, maps_to := β― }
β’ 0 < r / 2 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | mem_iff_eventually_subset | [11, 1] | [16, 51] | rw [(nhdsWithin_hasBasis nhds_basis_closedBall (Ioi (0 : β))).eventually_iff] | Ξ± : Type u_1
π : Type u_2
s : Set Ξ±
zβ : Ξ±
P : Ξ± β Prop
p : Filter Ξ±
Ο : β β Set Ξ±
hp : HasBasis p (fun t => 0 < t) Ο
hΟ : Monotone Ο
β’ s β p β βαΆ (t : β) in π[>] 0, Ο t β s | Ξ± : Type u_1
π : Type u_2
s : Set Ξ±
zβ : Ξ±
P : Ξ± β Prop
p : Filter Ξ±
Ο : β β Set Ξ±
hp : HasBasis p (fun t => 0 < t) Ο
hΟ : Monotone Ο
β’ s β p β β i, 0 < i β§ β β¦x : ββ¦, x β closedBall 0 i β© Ioi 0 β Ο x β s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | mem_iff_eventually_subset | [11, 1] | [16, 51] | simp_rw [hp.mem_iff, β exists_prop, mem_inter_iff, mem_closedBall_zero_iff] | Ξ± : Type u_1
π : Type u_2
s : Set Ξ±
zβ : Ξ±
P : Ξ± β Prop
p : Filter Ξ±
Ο : β β Set Ξ±
hp : HasBasis p (fun t => 0 < t) Ο
hΟ : Monotone Ο
β’ s β p β β i, 0 < i β§ β β¦x : ββ¦, x β closedBall 0 i β© Ioi 0 β Ο x β s | Ξ± : Type u_1
π : Type u_2
s : Set Ξ±
zβ : Ξ±
P : Ξ± β Prop
p : Filter Ξ±
Ο : β β Set Ξ±
hp : HasBasis p (fun t => 0 < t) Ο
hΟ : Monotone Ο
β’ (β i, β (_ : 0 < i), Ο i β s) β β i, β (_ : 0 < i), β β¦x : ββ¦, βxβ β€ i β§ x β Ioi 0 β Ο x β s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | mem_iff_eventually_subset | [11, 1] | [16, 51] | refine existsβ_congr (Ξ» Ξ΅ hΞ΅ => β¨Ξ» h r h' => (hΟ (le_of_abs_le h'.1)).trans h,
Ξ» h => h β¨Eq.le (abs_eq_self.mpr hΞ΅.le), hΞ΅β©β©) | Ξ± : Type u_1
π : Type u_2
s : Set Ξ±
zβ : Ξ±
P : Ξ± β Prop
p : Filter Ξ±
Ο : β β Set Ξ±
hp : HasBasis p (fun t => 0 < t) Ο
hΟ : Monotone Ο
β’ (β i, β (_ : 0 < i), Ο i β s) β β i, β (_ : 0 < i), β β¦x : ββ¦, βxβ β€ i β§ x β Ioi 0 β Ο x β s | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | dist_inv_le_dist_div | [33, 1] | [39, 9] | have h1 : x β 0 := by contrapose! hx; simp only [hx, mem_ball_self, hΞ·] | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hx : x β ball 0 Ξ·
hy : y β ball 0 Ξ·'
β’ dist xβ»ΒΉ yβ»ΒΉ β€ dist x y / (Ξ· * Ξ·') | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hx : x β ball 0 Ξ·
hy : y β ball 0 Ξ·'
h1 : x β 0
β’ dist xβ»ΒΉ yβ»ΒΉ β€ dist x y / (Ξ· * Ξ·') |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | dist_inv_le_dist_div | [33, 1] | [39, 9] | have h2 : y β 0 := by contrapose! hy; simp only [hy, mem_ball_self, hΞ·'] | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hx : x β ball 0 Ξ·
hy : y β ball 0 Ξ·'
h1 : x β 0
β’ dist xβ»ΒΉ yβ»ΒΉ β€ dist x y / (Ξ· * Ξ·') | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hx : x β ball 0 Ξ·
hy : y β ball 0 Ξ·'
h1 : x β 0
h2 : y β 0
β’ dist xβ»ΒΉ yβ»ΒΉ β€ dist x y / (Ξ· * Ξ·') |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | dist_inv_le_dist_div | [33, 1] | [39, 9] | simp only [mem_ball, dist_eq_norm, sub_zero, not_lt] at hx hy | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hx : x β ball 0 Ξ·
hy : y β ball 0 Ξ·'
h1 : x β 0
h2 : y β 0
β’ dist xβ»ΒΉ yβ»ΒΉ β€ dist x y / (Ξ· * Ξ·') | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
h1 : x β 0
h2 : y β 0
hx : Ξ· β€ βxβ
hy : Ξ·' β€ βyβ
β’ dist xβ»ΒΉ yβ»ΒΉ β€ dist x y / (Ξ· * Ξ·') |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | dist_inv_le_dist_div | [33, 1] | [39, 9] | rw [dist_inv_invβ h1 h2] | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
h1 : x β 0
h2 : y β 0
hx : Ξ· β€ βxβ
hy : Ξ·' β€ βyβ
β’ dist xβ»ΒΉ yβ»ΒΉ β€ dist x y / (Ξ· * Ξ·') | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
h1 : x β 0
h2 : y β 0
hx : Ξ· β€ βxβ
hy : Ξ·' β€ βyβ
β’ dist x y / (βxβ * βyβ) β€ dist x y / (Ξ· * Ξ·') |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | dist_inv_le_dist_div | [33, 1] | [39, 9] | gcongr | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
h1 : x β 0
h2 : y β 0
hx : Ξ· β€ βxβ
hy : Ξ·' β€ βyβ
β’ dist x y / (βxβ * βyβ) β€ dist x y / (Ξ· * Ξ·') | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | dist_inv_le_dist_div | [33, 1] | [39, 9] | contrapose! hx | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hx : x β ball 0 Ξ·
hy : y β ball 0 Ξ·'
β’ x β 0 | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hy : y β ball 0 Ξ·'
hx : x = 0
β’ x β ball 0 Ξ· |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | dist_inv_le_dist_div | [33, 1] | [39, 9] | simp only [hx, mem_ball_self, hΞ·] | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hy : y β ball 0 Ξ·'
hx : x = 0
β’ x β ball 0 Ξ· | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | dist_inv_le_dist_div | [33, 1] | [39, 9] | contrapose! hy | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hx : x β ball 0 Ξ·
hy : y β ball 0 Ξ·'
h1 : x β 0
β’ y β 0 | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hx : x β ball 0 Ξ·
h1 : x β 0
hy : y = 0
β’ y β ball 0 Ξ·' |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | dist_inv_le_dist_div | [33, 1] | [39, 9] | simp only [hy, mem_ball_self, hΞ·'] | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hΞ· : 0 < Ξ·
hΞ·' : 0 < Ξ·'
hx : x β ball 0 Ξ·
h1 : x β 0
hy : y = 0
β’ y β ball 0 Ξ·' | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | obtain β¨U, hU, V, hV, hUVβ© := inf_eq_bot_iff.mp hp | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π | case intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hV : V β π 0
hUV : U β© V = β
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | obtain β¨U', hU', V', hV', hUV'β© := inf_eq_bot_iff.mp hq | case intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hV : V β π 0
hUV : U β© V = β
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π | case intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hV : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV' : V' β π 0
hUV' : U' β© V' = β
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | obtain β¨Ξ·, hΞ·, hVβ© := Metric.mem_nhds_iff.mp hV | case intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hV : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV' : V' β π 0
hUV' : U' β© V' = β
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·' : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV' : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | obtain β¨Ξ·', hΞ·', hV'β© := Metric.mem_nhds_iff.mp hV' | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·' : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV' : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | have hΞ·Ξ·' : 0 < Ξ· * Ξ·' := mul_pos hΞ· hΞ·' | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | intro u hu | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
β’ map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) β€ π€ π | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
hu : u β π€ π
β’ u β map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | obtain β¨Ξ΅, hΞ΅, huβ© := mem_uniformity_dist.mp hu | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
hu : u β π€ π
β’ u β map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
β’ u β map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | rw [mem_map_iff_exists_image] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
β’ u β map (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) (π€ π β p ΓΛ’ q) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
β’ β s β π€ π β p ΓΛ’ q, (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) '' s β u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | refine β¨_, inter_mem_inf (dist_mem_uniformity (mul_pos hΞ΅ hΞ·Ξ·')) (prod_mem_prod hU hU'), ?_β© | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
β’ β s β π€ π β p ΓΛ’ q, (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) '' s β u | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
β’ (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) '' ({p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')} β© U ΓΛ’ U') β u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | rintro z β¨x, β¨hx1, hx2β©, rflβ© | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
β’ (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) '' ({p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')} β© U ΓΛ’ U') β u | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
x : π Γ π
hx1 : x β {p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')}
hx2 : x β U ΓΛ’ U'
β’ (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) x β u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | have hx'1 : x.1 β ball (0 : π) Ξ· :=
Ξ» h => (Set.nonempty_of_mem (mem_inter hx2.1 (hV h))).ne_empty hUV | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
x : π Γ π
hx1 : x β {p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')}
hx2 : x β U ΓΛ’ U'
β’ (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) x β u | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
x : π Γ π
hx1 : x β {p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')}
hx2 : x β U ΓΛ’ U'
hx'1 : x.1 β ball 0 Ξ·
β’ (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) x β u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | have hx'2 : x.2 β ball (0 : π) Ξ·' :=
Ξ» h => (Set.nonempty_of_mem (mem_inter hx2.2 (hV' h))).ne_empty hUV' | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
x : π Γ π
hx1 : x β {p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')}
hx2 : x β U ΓΛ’ U'
hx'1 : x.1 β ball 0 Ξ·
β’ (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) x β u | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
x : π Γ π
hx1 : x β {p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')}
hx2 : x β U ΓΛ’ U'
hx'1 : x.1 β ball 0 Ξ·
hx'2 : x.2 β ball 0 Ξ·'
β’ (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) x β u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | refine hu ((dist_inv_le_dist_div hΞ· hΞ·' hx'1 hx'2).trans_lt ?_) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
x : π Γ π
hx1 : x β {p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')}
hx2 : x β U ΓΛ’ U'
hx'1 : x.1 β ball 0 Ξ·
hx'2 : x.2 β ball 0 Ξ·'
β’ (fun x => (x.1β»ΒΉ, x.2β»ΒΉ)) x β u | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
x : π Γ π
hx1 : x β {p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')}
hx2 : x β U ΓΛ’ U'
hx'1 : x.1 β ball 0 Ξ·
hx'2 : x.2 β ball 0 Ξ·'
β’ dist x.1 x.2 / (Ξ· * Ξ·') < Ξ΅ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | convert (div_lt_div_right hΞ·Ξ·').mpr hx1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
x : π Γ π
hx1 : x β {p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')}
hx2 : x β U ΓΛ’ U'
hx'1 : x.1 β ball 0 Ξ·
hx'2 : x.2 β ball 0 Ξ·'
β’ dist x.1 x.2 / (Ξ· * Ξ·') < Ξ΅ | case h.e'_4
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
x : π Γ π
hx1 : x β {p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')}
hx2 : x β U ΓΛ’ U'
hx'1 : x.1 β ball 0 Ξ·
hx'2 : x.2 β ball 0 Ξ·'
β’ Ξ΅ = Ξ΅ * (Ξ· * Ξ·') / (Ξ· * Ξ·') |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | titi | [41, 1] | [59, 45] | field_simp [hΞ·.lt.ne.symm, hΞ·'.lt.ne.symm] | case h.e'_4
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ·β Ξ·'β : β
pβ : Filter ΞΉ
mf mg : β
p q : Filter π
hp : p β π 0 = β₯
hq : q β π 0 = β₯
U : Set π
hU : U β p
V : Set π
hVβ : V β π 0
hUV : U β© V = β
U' : Set π
hU' : U' β q
V' : Set π
hV'β : V' β π 0
hUV' : U' β© V' = β
Ξ· : β
hΞ· : Ξ· > 0
hV : ball 0 Ξ· β V
Ξ·' : β
hΞ·' : Ξ·' > 0
hV' : ball 0 Ξ·' β V'
hΞ·Ξ·' : 0 < Ξ· * Ξ·'
u : Set (π Γ π)
huβ : u β π€ π
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
hu : β {a b : π}, dist a b < Ξ΅ β (a, b) β u
x : π Γ π
hx1 : x β {p | dist p.1 p.2 < Ξ΅ * (Ξ· * Ξ·')}
hx2 : x β U ΓΛ’ U'
hx'1 : x.1 β ball 0 Ξ·
hx'2 : x.2 β ball 0 Ξ·'
β’ Ξ΅ = Ξ΅ * (Ξ· * Ξ·') / (Ξ· * Ξ·') | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | uniform_ContinuousOn_inv | [61, 1] | [63, 89] | simpa only [UniformContinuousOn, Tendsto, β prod_principal_principal] using titi hs hs | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
sβ K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
s : Set π
hs : π s β π 0 = β₯
β’ UniformContinuousOn Inv.inv s | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.inv | [65, 1] | [72, 63] | have : πα΅ (f '' s) β π 0 = β₯ := by
rw [inf_comm] at hf β’
exact UniformSpace.nhds_inf_uniform_nhds_eq_bot hf | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hf : π (f '' s) β π 0 = β₯
β’ TendstoUniformlyOn Fβ»ΒΉ fβ»ΒΉ p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hf : π (f '' s) β π 0 = β₯
this : πα΅ (f '' s) β π 0 = β₯
β’ TendstoUniformlyOn Fβ»ΒΉ fβ»ΒΉ p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.inv | [65, 1] | [72, 63] | have h1 := lemma1 hF | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hf : π (f '' s) β π 0 = β₯
this : πα΅ (f '' s) β π 0 = β₯
β’ TendstoUniformlyOn Fβ»ΒΉ fβ»ΒΉ p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hf : π (f '' s) β π 0 = β₯
this : πα΅ (f '' s) β π 0 = β₯
h1 : Tendsto (fun q => (f q.2, F q.1 q.2)) (p ΓΛ’ π s) (π (f '' s) ΓΛ’ πα΅ (f '' s))
β’ TendstoUniformlyOn Fβ»ΒΉ fβ»ΒΉ p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.inv | [65, 1] | [72, 63] | rw [tendstoUniformlyOn_iff_tendsto] at hF β’ | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hf : π (f '' s) β π 0 = β₯
this : πα΅ (f '' s) β π 0 = β₯
h1 : Tendsto (fun q => (f q.2, F q.1 q.2)) (p ΓΛ’ π s) (π (f '' s) ΓΛ’ πα΅ (f '' s))
β’ TendstoUniformlyOn Fβ»ΒΉ fβ»ΒΉ p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : Tendsto (fun q => (f q.2, F q.1 q.2)) (p ΓΛ’ π s) (π€ π)
hf : π (f '' s) β π 0 = β₯
this : πα΅ (f '' s) β π 0 = β₯
h1 : Tendsto (fun q => (f q.2, F q.1 q.2)) (p ΓΛ’ π s) (π (f '' s) ΓΛ’ πα΅ (f '' s))
β’ Tendsto (fun q => (fβ»ΒΉ q.2, Fβ»ΒΉ q.1 q.2)) (p ΓΛ’ π s) (π€ π) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.inv | [65, 1] | [72, 63] | refine (Filter.map_mono (le_inf hF h1)).trans (titi hf this) | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : Tendsto (fun q => (f q.2, F q.1 q.2)) (p ΓΛ’ π s) (π€ π)
hf : π (f '' s) β π 0 = β₯
this : πα΅ (f '' s) β π 0 = β₯
h1 : Tendsto (fun q => (f q.2, F q.1 q.2)) (p ΓΛ’ π s) (π (f '' s) ΓΛ’ πα΅ (f '' s))
β’ Tendsto (fun q => (fβ»ΒΉ q.2, Fβ»ΒΉ q.1 q.2)) (p ΓΛ’ π s) (π€ π) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.inv | [65, 1] | [72, 63] | rw [inf_comm] at hf β’ | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hf : π (f '' s) β π 0 = β₯
β’ πα΅ (f '' s) β π 0 = β₯ | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hf : π 0 β π (f '' s) = β₯
β’ π 0 β πα΅ (f '' s) = β₯ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.inv | [65, 1] | [72, 63] | exact UniformSpace.nhds_inf_uniform_nhds_eq_bot hf | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hf : π 0 β π (f '' s) = β₯
β’ π 0 β πα΅ (f '' s) = β₯ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | lxyab | [74, 1] | [74, 81] | ring | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ yβ : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
x y a b : π
β’ x * a - y * b = (x - y) * a + y * (a - b) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | by_cases h : NeBot p | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
β’ TendstoUniformlyOn (F * G) (f * g) p s | case pos
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
β’ TendstoUniformlyOn (F * G) (f * g) p s
case neg
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : Β¬NeBot p
β’ TendstoUniformlyOn (F * G) (f * g) p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | case neg => simp at h; simp [h, TendstoUniformlyOn] | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : Β¬NeBot p
β’ TendstoUniformlyOn (F * G) (f * g) p s | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | case pos =>
set Mf := |mf| + 1
set Mg := |mg| + 1
have hMf : 0 < Mf := by positivity
have hMg : 0 < Mg := by positivity
replace hf : βαΆ i in p, β x β s, βF i xβ β€ Mf := by
filter_upwards [hf] with i hF x hx using (hF x hx).trans ((le_abs_self mf).trans (lt_add_one _).le)
replace hg : βαΆ i in p, β x β s, βG i xβ β€ Mg := by
filter_upwards [hg] with i hG x hx using (hG x hx).trans ((le_abs_self mg).trans (lt_add_one _).le)
have h1 : β x β s, βg xβ β€ Mg := by
intro x hx
refine le_of_tendsto ((continuous_norm.tendsto (g x)).comp (hG.tendsto_at hx)) ?_
filter_upwards [hg] with i hg using hg x hx
simp_rw [Metric.tendstoUniformlyOn_iff, dist_eq_norm] at hF hG β’
intro Ξ΅ hΞ΅
filter_upwards [hf, hF (Ξ΅ / (2 * Mg)) (by positivity), hG (Ξ΅ / (2 * Mf)) (by positivity)] with i hf hF hG x hx
have h2 : β(f x - F i x) * g xβ < Ξ΅ / 2 := by
rw [norm_mul]
by_cases h : g x = 0
case pos => simp [h, half_pos hΞ΅]
case neg =>
convert mul_lt_mul (hF x hx) (h1 x hx) (norm_pos_iff.mpr h) (by positivity) using 1
simp only [div_mul, mul_div_cancel, hMg.ne.symm, Ne.def, not_false_iff]
have h3 : βF i x * (g x - G i x)β < Ξ΅ / 2 := by
rw [norm_mul]
by_cases h : F i x = 0
case pos => simp [h, half_pos hΞ΅]
case neg =>
convert mul_lt_mul' (hf x hx) (hG x hx) (norm_nonneg _) hMf using 1
field_simp [hMf.ne.symm]; ring
simp_rw [Pi.mul_apply, lxyab]
exact (norm_add_le _ _).trans_lt (add_halves' Ξ΅ βΈ add_lt_add h2 h3) | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
β’ TendstoUniformlyOn (F * G) (f * g) p s | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | simp at h | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : Β¬NeBot p
β’ TendstoUniformlyOn (F * G) (f * g) p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : p = β₯
β’ TendstoUniformlyOn (F * G) (f * g) p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | simp [h, TendstoUniformlyOn] | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : p = β₯
β’ TendstoUniformlyOn (F * G) (f * g) p s | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | set Mf := |mf| + 1 | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
β’ TendstoUniformlyOn (F * G) (f * g) p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
Mf : β := |mf| + 1
β’ TendstoUniformlyOn (F * G) (f * g) p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | set Mg := |mg| + 1 | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
Mf : β := |mf| + 1
β’ TendstoUniformlyOn (F * G) (f * g) p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
β’ TendstoUniformlyOn (F * G) (f * g) p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | have hMf : 0 < Mf := by positivity | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
β’ TendstoUniformlyOn (F * G) (f * g) p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
β’ TendstoUniformlyOn (F * G) (f * g) p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | have hMg : 0 < Mg := by positivity | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
β’ TendstoUniformlyOn (F * G) (f * g) p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
β’ TendstoUniformlyOn (F * G) (f * g) p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | replace hf : βαΆ i in p, β x β s, βF i xβ β€ Mf := by
filter_upwards [hf] with i hF x hx using (hF x hx).trans ((le_abs_self mf).trans (lt_add_one _).le) | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
β’ TendstoUniformlyOn (F * G) (f * g) p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
β’ TendstoUniformlyOn (F * G) (f * g) p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | replace hg : βαΆ i in p, β x β s, βG i xβ β€ Mg := by
filter_upwards [hg] with i hG x hx using (hG x hx).trans ((le_abs_self mg).trans (lt_add_one _).le) | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ mg
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
β’ TendstoUniformlyOn (F * G) (f * g) p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ Mg
β’ TendstoUniformlyOn (F * G) (f * g) p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | have h1 : β x β s, βg xβ β€ Mg := by
intro x hx
refine le_of_tendsto ((continuous_norm.tendsto (g x)).comp (hG.tendsto_at hx)) ?_
filter_upwards [hg] with i hg using hg x hx | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ Mg
β’ TendstoUniformlyOn (F * G) (f * g) p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ Mg
h1 : β x β s, βg xβ β€ Mg
β’ TendstoUniformlyOn (F * G) (f * g) p s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | simp_rw [Metric.tendstoUniformlyOn_iff, dist_eq_norm] at hF hG β’ | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ Mg
h1 : β x β s, βg xβ β€ Mg
β’ TendstoUniformlyOn (F * G) (f * g) p s | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ Mg
h1 : β x β s, βg xβ β€ Mg
hF : β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, βf x - F n xβ < Ξ΅
hG : β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, βg x - G n xβ < Ξ΅
β’ β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, β(f * g) x - (F * G) n xβ < Ξ΅ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | intro Ξ΅ hΞ΅ | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ Mg
h1 : β x β s, βg xβ β€ Mg
hF : β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, βf x - F n xβ < Ξ΅
hG : β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, βg x - G n xβ < Ξ΅
β’ β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, β(f * g) x - (F * G) n xβ < Ξ΅ | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ Mg
h1 : β x β s, βg xβ β€ Mg
hF : β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, βf x - F n xβ < Ξ΅
hG : β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, βg x - G n xβ < Ξ΅
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
β’ βαΆ (n : ΞΉ) in p, β x β s, β(f * g) x - (F * G) n xβ < Ξ΅ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | filter_upwards [hf, hF (Ξ΅ / (2 * Mg)) (by positivity), hG (Ξ΅ / (2 * Mf)) (by positivity)] with i hf hF hG x hx | π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
x y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ Mg
h1 : β x β s, βg xβ β€ Mg
hF : β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, βf x - F n xβ < Ξ΅
hG : β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, βg x - G n xβ < Ξ΅
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
β’ βαΆ (n : ΞΉ) in p, β x β s, β(f * g) x - (F * G) n xβ < Ξ΅ | case h
π : Type u_1
ΞΉ : Type u_2
Ξ± : Type u_3
s K : Set Ξ±
instβ : NormedField π
F G : ΞΉ β Ξ± β π
f g : Ξ± β π
xβ y : π
Ξ· Ξ·' : β
p : Filter ΞΉ
mf mg : β
h : NeBot p
Mf : β := |mf| + 1
Mg : β := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hfβ : βαΆ (i : ΞΉ) in p, β x β s, βF i xβ β€ Mf
hg : βαΆ (i : ΞΉ) in p, β x β s, βG i xβ β€ Mg
h1 : β x β s, βg xβ β€ Mg
hFβ : β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, βf x - F n xβ < Ξ΅
hGβ : β Ξ΅ > 0, βαΆ (n : ΞΉ) in p, β x β s, βg x - G n xβ < Ξ΅
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
i : ΞΉ
hf : β x β s, βF i xβ β€ Mf
hF : β x β s, βf x - F i xβ < Ξ΅ / (2 * Mg)
hG : β x β s, βg x - G i xβ < Ξ΅ / (2 * Mf)
x : Ξ±
hx : x β s
β’ β(f * g) x - (F * G) i xβ < Ξ΅ |
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