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stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
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stringlengths 1
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stringlengths 3
2.09M
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stringlengths 73
2.09M
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | clear GS G h | case h.intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | set A := N βͺ M \ N | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | have AM : M β A := subset_union_sdiff _ _ | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
AM : M β A
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | simp at HM | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
AM : M β A
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : β (n : Finset β), M β n β dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
AM : M β A
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | specialize HM A AM | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : β (n : Finset β), M β n β dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist (A.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : β (n : Finset β), M β n β dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [dist_comm] at HM | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist (A.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist (A.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | calc dist g (N.sum fun n β¦ f n z)
_ β€ dist g (A.sum fun n β¦ f n z) + dist (A.sum fun n β¦ f n z) (N.sum fun n β¦ f n z) := by bound
_ β€ e / 4 + dist (A.sum fun n β¦ f n z) (N.sum fun n β¦ f n z) := by linarith
_ = e / 4 + dist ((N.sum fun n β¦ f n z) + (M \ N).sum fun n β¦ f n z)
(N.sum fun n β¦ f n z) := by rw [Finset.sum_union Finset.disjoint_sdiff]
_ = e / 4 + abs (((N.sum fun n β¦ f n z) + (M \ N).sum fun n β¦ f n z) -
N.sum fun n β¦ f n z) := by rw [Complex.dist_eq]
_ = e / 4 + abs ((M \ N).sum fun n β¦ f n z) := by ring_nf
_ β€ e / 4 + (M \ N).sum fun n β¦ abs (f n z) := by
linarith [finset_complex_abs_sum_le (M \ N) fun n β¦ f n z]
_ β€ e / 4 + e / 4 := by linarith [hm (M \ N) z (sdiff_late M Nm) zs]
_ = e / 2 := by ring
_ < e := by linarith | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (N.sum fun n => f n z) < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
β’ e / 4 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
β’ e / 4 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [β G] | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ HasSum (fun n => f n z) g | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ HasSum (fun n => f n z) (β' (n : β), f n z) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ HasSum (fun n => f n z) g
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | exact Summable.hasSum S | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ HasSum (fun n => f n z) (β' (n : β), f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ HasSum (fun n => f n z) (β' (n : β), f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
β’ e / 4 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
β’ e / 4 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | bound | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (N.sum fun n => f n z) β€ dist g (A.sum fun n => f n z) + dist (A.sum fun n => f n z) (N.sum fun n => f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (N.sum fun n => f n z) β€ dist g (A.sum fun n => f n z) + dist (A.sum fun n => f n z) (N.sum fun n => f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (A.sum fun n => f n z) + dist (A.sum fun n => f n z) (N.sum fun n => f n z) β€
e / 4 + dist (A.sum fun n => f n z) (N.sum fun n => f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (A.sum fun n => f n z) + dist (A.sum fun n => f n z) (N.sum fun n => f n z) β€
e / 4 + dist (A.sum fun n => f n z) (N.sum fun n => f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [Finset.sum_union Finset.disjoint_sdiff] | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + dist (A.sum fun n => f n z) (N.sum fun n => f n z) =
e / 4 + dist ((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) (N.sum fun n => f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + dist (A.sum fun n => f n z) (N.sum fun n => f n z) =
e / 4 + dist ((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) (N.sum fun n => f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [Complex.dist_eq] | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + dist ((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) (N.sum fun n => f n z) =
e / 4 + Complex.abs (((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) - N.sum fun n => f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + dist ((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) (N.sum fun n => f n z) =
e / 4 + Complex.abs (((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) - N.sum fun n => f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | ring_nf | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + Complex.abs (((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) - N.sum fun n => f n z) =
e / 4 + Complex.abs ((M \ N).sum fun n => f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + Complex.abs (((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) - N.sum fun n => f n z) =
e / 4 + Complex.abs ((M \ N).sum fun n => f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith [finset_complex_abs_sum_le (M \ N) fun n β¦ f n z] | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + Complex.abs ((M \ N).sum fun n => f n z) β€ e / 4 + (M \ N).sum fun n => Complex.abs (f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + Complex.abs ((M \ N).sum fun n => f n z) β€ e / 4 + (M \ N).sum fun n => Complex.abs (f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith [hm (M \ N) z (sdiff_late M Nm) zs] | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ (e / 4 + (M \ N).sum fun n => Complex.abs (f n z)) β€ e / 4 + e / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ (e / 4 + (M \ N).sum fun n => Complex.abs (f n z)) β€ e / 4 + e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | ring | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + e / 4 = e / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + e / 4 = e / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 2 < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 2 < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | intro n z zs | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ β (n : β), β z β s, f n z = 0 | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
n : β
z : β
zs : z β s
β’ f n z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ β (n : β), β z β s, f n z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | specialize hf n z zs | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
n : β
z : β
zs : z β s
β’ f n z = 0 | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
β’ f n z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
n : β
z : β
zs : z β s
β’ f n z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | have ca : c * a ^ n β€ 0 := mul_nonpos_iff.mpr (Or.inr β¨c0, by boundβ©) | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
β’ f n z = 0 | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
ca : c * a ^ n β€ 0
β’ f n z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
β’ f n z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | exact Complex.abs.eq_zero.mp (le_antisymm (le_trans hf ca) (Complex.abs.nonneg _)) | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
ca : c * a ^ n β€ 0
β’ f n z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
ca : c * a ^ n β€ 0
β’ f n z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | bound | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
β’ 0 β€ a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
β’ 0 β€ a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | by_cases c0 : c β€ 0 | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ HasUniformSum f (tsumOn f) s | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
β’ HasUniformSum f (tsumOn f) s
case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : Β¬c β€ 0
β’ HasUniformSum f (tsumOn f) s | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | have fz := CNonpos.degenerate c0 a0 hf | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
β’ HasUniformSum f (tsumOn f) s | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ HasUniformSum f (tsumOn f) s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rw [HasUniformSum, Metric.tendstoUniformlyOn_iff] | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ HasUniformSum f (tsumOn f) s | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ β Ξ΅ > 0, βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro e ep | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ β Ξ΅ > 0, βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < Ξ΅ | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ β Ξ΅ > 0, βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < Ξ΅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | apply Filter.eventually_of_forall | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ β (x : Finset β), β x_1 β s, dist (tsumOn f x_1) (x.sum fun n => f n x_1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro n z zs | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ β (x : Finset β), β x_1 β s, dist (tsumOn f x_1) (x.sum fun n => f n x_1) < e | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (tsumOn f z) (n.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ β (x : Finset β), β x_1 β s, dist (tsumOn f x_1) (x.sum fun n => f n x_1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rw [tsumOn] | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (tsumOn f z) (n.sum fun n => f n z) < e | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), f n z) (n.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (tsumOn f z) (n.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | simp_rw [fz _ z zs] | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), f n z) (n.sum fun n => f n z) < e | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), 0) (n.sum fun n => 0) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), f n z) (n.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | simp only [tsum_zero, Finset.sum_const_zero, dist_zero_left, Complex.norm_eq_abs,
AbsoluteValue.map_zero] | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), 0) (n.sum fun n => 0) < e | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ 0 < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), 0) (n.sum fun n => 0) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | assumption | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ 0 < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ 0 < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | simp only [not_le] at c0 | case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : Β¬c β€ 0
β’ HasUniformSum f (tsumOn f) s | case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ HasUniformSum f (tsumOn f) s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : Β¬c β€ 0
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | apply uniformVanishing_to_tendsto_uniformly_on | case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ HasUniformSum f (tsumOn f) s | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ UniformVanishing f s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro e ep | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ UniformVanishing f s | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ UniformVanishing f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | set t := (1 - βa) / βc * (e / 2) | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | have tp : t > 0 := by bound | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rcases exists_pow_lt_of_lt_one tp a1 with β¨n, ntβ© | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | case neg.h.intro
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | use n | case neg.h.intro
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro N z NL zs | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
β’ (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | have a1p : 1 - (a : β) > 0 := by linarith | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
β’ (N.sum fun n => Complex.abs (f n z)) < e | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
β’ (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | calc (N.sum fun n β¦ abs (f n z))
_ β€ N.sum fun n β¦ c * a ^ n := Finset.sum_le_sum fun n _ β¦ hf n z zs
_ = c * N.sum fun n β¦ a ^ n := (Finset.mul_sum _ _ _).symm
_ β€ c * (a ^ n * (1 - a)β»ΒΉ) := by bound [late_geometric_bound NL a0 a1]
_ = a ^ n * (c * (1 - a)β»ΒΉ) := by ring
_ β€ t * (c * (1 - a)β»ΒΉ) := by bound
_ = (1 - a) / c * (e / 2) * (c * (1 - a)β»ΒΉ) := rfl
_ = (1 - a) * (1 - a)β»ΒΉ * (c / c) * (e / 2) := by ring
_ = 1 * 1 * (e / 2) := by rw [mul_inv_cancel a1p.ne', div_self c0.ne']
_ = e / 2 := by ring
_ < e := by linarith | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (N.sum fun n => Complex.abs (f n z)) < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | bound | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
β’ t > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
β’ t > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | linarith | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
β’ 1 - a > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
β’ 1 - a > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | bound [late_geometric_bound NL a0 a1] | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (c * N.sum fun n => a ^ n) β€ c * (a ^ n * (1 - a)β»ΒΉ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (c * N.sum fun n => a ^ n) β€ c * (a ^ n * (1 - a)β»ΒΉ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | ring | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ c * (a ^ n * (1 - a)β»ΒΉ) = a ^ n * (c * (1 - a)β»ΒΉ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ c * (a ^ n * (1 - a)β»ΒΉ) = a ^ n * (c * (1 - a)β»ΒΉ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | bound | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ a ^ n * (c * (1 - a)β»ΒΉ) β€ t * (c * (1 - a)β»ΒΉ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ a ^ n * (c * (1 - a)β»ΒΉ) β€ t * (c * (1 - a)β»ΒΉ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | ring | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (1 - a) / c * (e / 2) * (c * (1 - a)β»ΒΉ) = (1 - a) * (1 - a)β»ΒΉ * (c / c) * (e / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (1 - a) / c * (e / 2) * (c * (1 - a)β»ΒΉ) = (1 - a) * (1 - a)β»ΒΉ * (c / c) * (e / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rw [mul_inv_cancel a1p.ne', div_self c0.ne'] | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (1 - a) * (1 - a)β»ΒΉ * (c / c) * (e / 2) = 1 * 1 * (e / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (1 - a) * (1 - a)β»ΒΉ * (c / c) * (e / 2) = 1 * 1 * (e / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | ring | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ 1 * 1 * (e / 2) = e / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ 1 * 1 * (e / 2) = e / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | linarith | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ e / 2 < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ e / 2 < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | set s : Set β := {0} | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | set g : β β β β β := fun n _ β¦ f n | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | have hg : β n z, z β s β abs (g n z) β€ c * a ^ n := fun n z _ β¦ hf n | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | have u := fast_series_converge_uniformly_on a0 a1 hg | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : HasUniformSum (fun n z => g n z) (tsumOn fun n z => g n z) s
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | rw [HasUniformSum] at u | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : HasUniformSum (fun n z => g n z) (tsumOn fun n z => g n z) s
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : TendstoUniformlyOn (fun N z => N.sum fun n => g n z) (tsumOn fun n z => g n z) atTop s
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : HasUniformSum (fun n z => g n z) (tsumOn fun n z => g n z) s
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | rw [tendstoUniformlyOn_singleton_iff_tendsto] at u | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : TendstoUniformlyOn (fun N z => N.sum fun n => g n z) (tsumOn fun n z => g n z) atTop s
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : TendstoUniformlyOn (fun N z => N.sum fun n => g n z) (tsumOn fun n z => g n z) atTop s
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | apply HasSum.summable | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ Summable f | case h
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ HasSum f ?a
case a
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | assumption | case h
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ HasSum f ?a
case a
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ HasSum f ?a
case a
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | use tsumOn f | f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ β g, AnalyticOn β g s β§ HasSumOn f g s | case h
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s β§ HasSumOn f (tsumOn f) s | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ β g, AnalyticOn β g s β§ HasSumOn f g s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | constructor | case h
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s β§ HasSumOn f (tsumOn f) s | case h.left
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s
case h.right
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ HasSumOn f (tsumOn f) s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s β§ HasSumOn f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | exact uniform_analytic_lim o (fun N β¦ N.analyticOn_sum fun _ _ β¦ h _)
(fast_series_converge_uniformly_on a0 a1 hf) | case h.left
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | exact fun z zs β¦ Summable.hasSum (fast_series_converge_at a0 a1 fun n β¦ hf n z zs) | case h.right
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ HasSumOn f (tsumOn f) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ HasSumOn f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | rw [HasSum] at h β’ | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : HasSum f g
β’ HasSum (Stream'.cons a f) (a + g) | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : HasSum f g
β’ HasSum (Stream'.cons a f) (a + g)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | have ha := Filter.Tendsto.comp (Continuous.tendsto (continuous_add_left a) g) h | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | have s : ((fun z β¦ a + z) β fun N : Finset β β¦ N.sum f) =
(fun N : Finset β β¦ N.sum (Stream'.cons a f)) β push := by
apply funext; intro N; simp; exact push_sum | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | rw [s] at ha | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun N => N.sum (Stream'.cons a f)) β push) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | exact tendsto_comp_push.mp ha | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun N => N.sum (Stream'.cons a f)) β push) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun N => N.sum (Stream'.cons a f)) β push) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | apply funext | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ β (x : Finset β), ((fun z => a + z) β fun N => N.sum f) x = ((fun N => N.sum (Stream'.cons a f)) β push) x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | intro N | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ β (x : Finset β), ((fun z => a + z) β fun N => N.sum f) x = ((fun N => N.sum (Stream'.cons a f)) β push) x | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ ((fun z => a + z) β fun N => N.sum f) N = ((fun N => N.sum (Stream'.cons a f)) β push) N | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ β (x : Finset β), ((fun z => a + z) β fun N => N.sum f) x = ((fun N => N.sum (Stream'.cons a f)) β push) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | simp | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ ((fun z => a + z) β fun N => N.sum f) N = ((fun N => N.sum (Stream'.cons a f)) β push) N | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ a + N.sum f = (push N).sum (Stream'.cons a f) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ ((fun z => a + z) β fun N => N.sum f) N = ((fun N => N.sum (Stream'.cons a f)) β push) N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | exact push_sum | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ a + N.sum f = (push N).sum (Stream'.cons a f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ a + N.sum f = (push N).sum (Stream'.cons a f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | rcases h with β¨g, hβ© | a : β
f : β β β
h : Summable f
β’ tsum (Stream'.cons a f) = a + tsum f | case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + tsum f | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
f : β β β
h : Summable f
β’ tsum (Stream'.cons a f) = a + tsum f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | rw [HasSum.tsum_eq h] | case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + tsum f | case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + g | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + tsum f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | rw [HasSum.tsum_eq _] | case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + g | a : β
f : β β β
g : β
h : HasSum f g
β’ HasSum (fun b => Stream'.cons a f b) (a + g) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + g
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | exact sum_cons h | a : β
f : β β β
g : β
h : HasSum f g
β’ HasSum (fun b => Stream'.cons a f b) (a + g) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
f : β β β
g : β
h : HasSum f g
β’ HasSum (fun b => Stream'.cons a f b) (a + g)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | have c := sum_cons (a := -f 0) h | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
β’ HasSum (fun n => f (n + 1)) (g - f 0) | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ HasSum (fun n => f (n + 1)) (g - f 0) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
β’ HasSum (fun n => f (n + 1)) (g - f 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | rw [HasSum] | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ HasSum (fun n => f (n + 1)) (g - f 0) | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ HasSum (fun n => f (n + 1)) (g - f 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | rw [neg_add_eq_sub, HasSum, β tendsto_comp_push, β tendsto_comp_push] at c | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | have s : ((fun N : Finset β β¦ N.sum fun n β¦ (Stream'.cons (-f 0) f) n) β push) β push =
fun N : Finset β β¦ N.sum fun n β¦ f (n + 1) := by
clear c h g; apply funext; intro N; simp
nth_rw 2 [β Stream'.eta f]
simp only [βpush_sum, Stream'.head, Stream'.tail, Stream'.get]
abel | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | rw [s] at c | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (fun N => N.sum fun n => f (n + 1)) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | assumption | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (fun N => N.sum fun n => f (n + 1)) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (fun N => N.sum fun n => f (n + 1)) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | clear c h g | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
β’ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1) | X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
β’ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | apply funext | X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1) | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ β (x : Finset β), (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) x = x.sum fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | intro N | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ β (x : Finset β), (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) x = x.sum fun n => f (n + 1) | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) N = N.sum fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ β (x : Finset β), (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) x = x.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | simp | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) N = N.sum fun n => f (n + 1) | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) f n) = N.sum fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) N = N.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | nth_rw 2 [β Stream'.eta f] | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) f n) = N.sum fun n => f (n + 1) | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) (Stream'.cons (Stream'.head f) (Stream'.tail f)) n) =
N.sum fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) f n) = N.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | simp only [βpush_sum, Stream'.head, Stream'.tail, Stream'.get] | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) (Stream'.cons (Stream'.head f) (Stream'.tail f)) n) =
N.sum fun n => f (n + 1) | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ -f 0 + (f 0 + N.sum fun x => f (x + 1)) = N.sum fun x => f (x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) (Stream'.cons (Stream'.head f) (Stream'.tail f)) n) =
N.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | abel | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ -f 0 + (f 0 + N.sum fun x => f (x + 1)) = N.sum fun x => f (x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ -f 0 + (f 0 + N.sum fun x => f (x + 1)) = N.sum fun x => f (x + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_mono | [21, 1] | [23, 73] | positivity | c : β
d : β
instβ : Fact (2 β€ d)
x y : β
x0 : 1 < x
xy : x β€ y
β’ 0 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x y : β
x0 : 1 < x
xy : x β€ y
β’ 0 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_neg_log_strict_anti | [25, 1] | [30, 76] | have lx := neg_pos.mpr (Real.log_neg x0 x1) | c : β
d : β
instβ : Fact (2 β€ d)
x y : β
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
β’ (-y.log).log < (-x.log).log β x < y | c : β
d : β
instβ : Fact (2 β€ d)
x y : β
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
β’ (-y.log).log < (-x.log).log β x < y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x y : β
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
β’ (-y.log).log < (-x.log).log β x < y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_neg_log_strict_anti | [25, 1] | [30, 76] | have ly := neg_pos.mpr (Real.log_neg y0 y1) | c : β
d : β
instβ : Fact (2 β€ d)
x y : β
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
β’ (-y.log).log < (-x.log).log β x < y | c : β
d : β
instβ : Fact (2 β€ d)
x y : β
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
ly : 0 < -y.log
β’ (-y.log).log < (-x.log).log β x < y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x y : β
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
β’ (-y.log).log < (-x.log).log β x < y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_neg_log_strict_anti | [25, 1] | [30, 76] | rw [Real.log_lt_log_iff ly lx, neg_lt_neg_iff, Real.log_lt_log_iff x0 y0] | c : β
d : β
instβ : Fact (2 β€ d)
x y : β
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
ly : 0 < -y.log
β’ (-y.log).log < (-x.log).log β x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x y : β
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
ly : 0 < -y.log
β’ (-y.log).log < (-x.log).log β x < y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | rw [Real.le_log_iff_exp_le (by linarith)] | c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
β’ log 2 * x β€ (1 + x).log | c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
β’ (log 2 * x).exp β€ 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
β’ log 2 * x β€ (1 + x).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | have x0' : 0 β€ 1 - x := by linarith | c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
β’ (log 2 * x).exp β€ 1 + x | c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
x0' : 0 β€ 1 - x
β’ (log 2 * x).exp β€ 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
β’ (log 2 * x).exp β€ 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | have h := convexOn_exp.2 (mem_univ 0) (mem_univ (log 2)) x0' x0 (by abel) | c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
x0' : 0 β€ 1 - x
β’ (log 2 * x).exp β€ 1 + x | c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
x0' : 0 β€ 1 - x
h : ((1 - x) β’ 0 + x β’ log 2).exp β€ (1 - x) β’ exp 0 + x β’ (log 2).exp
β’ (log 2 * x).exp β€ 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
x0' : 0 β€ 1 - x
β’ (log 2 * x).exp β€ 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | simp only [smul_eq_mul, mul_zero, zero_add, Real.exp_zero, mul_one, Real.exp_log zero_lt_two] at h | c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
x0' : 0 β€ 1 - x
h : ((1 - x) β’ 0 + x β’ log 2).exp β€ (1 - x) β’ exp 0 + x β’ (log 2).exp
β’ (log 2 * x).exp β€ 1 + x | c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
x0' : 0 β€ 1 - x
h : (x * log 2).exp β€ 1 - x + x * 2
β’ (log 2 * x).exp β€ 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x : β
x0 : 0 β€ x
x1 : x β€ 1
x0' : 0 β€ 1 - x
h : ((1 - x) β’ 0 + x β’ log 2).exp β€ (1 - x) β’ exp 0 + x β’ (log 2).exp
β’ (log 2 * x).exp β€ 1 + x
TACTIC:
|
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